Adiabatic quenches and characterization of amplitudeexcitations in a continuous quantum phase transitionThai M. Hoanga, Hebbe M. Bharatha, Matthew J. Boguslawskia, Martin Anqueza, Bryce A. Robbinsa,and Michael S. Chapmana,1

aSchool of Physics, Georgia Institute of Technology, Atlanta, GA 30332-0430

Edited by Subir Sachdev, Harvard University, Cambridge, MA, and approved June 28, 2016 (received for review January 7, 2016)

Spontaneous symmetry breaking occurs in a physical systemwhenever the ground state does not share the symmetry of theunderlying theory, e.g., the Hamiltonian. This mechanism gives riseto massless Nambu–Goldstone modes and massive Anderson–Higgs modes. These modes provide a fundamental understandingof matter in the Universe and appear as collective phase or ampli-tude excitations of an order parameter in a many-body system. Theamplitude excitation plays a crucial role in determining the criticalexponents governing universal nonequilibrium dynamics in theKibble–Zurek mechanism (KZM). Here, we characterize the ampli-tude excitations in a spin-1 condensate and measure the energygap for different phases of the quantum phase transition. At thequantum critical point of the transition, finite-size effects lead to anonzero gap. Our measurements are consistent with this predic-tion, and furthermore, we demonstrate an adiabatic quenchthrough the phase transition, which is forbidden at the mean fieldlevel. This work paves the way toward generating entanglementthrough an adiabatic phase transition.

adiabatic quenches | amplitude excitations | quantum phase transition

The amplitude mode and phase mode describe two distinctexcitation degrees of freedom of a complex order parameter

ψ =Aeiϕ appearing in many quantum systems such as the orderparameter of the Ginzburg–Laudau superconducting phase tran-sition (1) and the two-component quantum field of the Nambu–Goldstone–Anderson–Higgs matter field model (2–5). In a zero-dimensional system of an interacting spin-1 condensate, thetransverse spin, S⊥, plays the role of an order parameter in thequantum phase transition (QPT) with S⊥ being zero in the polar(P) phase and nonzero in the broken axisymmetry (BA) phase(Fig. 1A). Representing the transverse spin vector as a complexnumber, S⊥ = Sx + iSy, with the real and imaginary parts beingexpectation values of spin-1 operators, the amplitude modecorresponds to the amplitude oscillation of S⊥.The amplitude mode can be studied in different spinor phases

by tuning the relative strengths of the quadratic Zeeman energyper particle q∝B2 and spin interaction energy c of the conden-sate (6) by varying the magnetic field strength B (Fig. 1). In theP phase, both the effective spinor potential energy V and theground state (GS) spin vector have SO(2) rotational symmetryabout the vertical axis (Fig. 1A), and there are two degeneratecollective amplitude modes along the radial directions about theGS located at the bottom of the parabolic bowl. These amplitudeexcitations are gapped modes, which vary both the amplitude ofS⊥ and the energy.In the BA phase, the effective spinor potential energy V ac-

quires a Mexican-hat shape with the GS occupying the minimalenergy ring of radius

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4c2 − q2

p=ð2jcjÞ. The GS spin vector, S⊥

(orange arrow in Fig. 1A), spontaneously breaks the SO(2)symmetry and acquires a definite direction (7, 8). This brokensymmetry induces a massless Nambu–Goldstone (NG) mode inwhich it costs no energy for the spin vector to rotate about thevertical axis. Recently, the magnetic dipolar interaction wasused to open a gap in the NG mode by breaking the rotationalsymmetry of the spin interaction (9). In our condensate, the

magnetic dipolar interaction can be ignored due to spatialisotropy, and therefore, the NG mode in the BA phase remainsgapless. The other excitation, the amplitude mode, manifestsitself as an amplitude oscillation of the transverse spin in theradial direction. This amplitude mode is similar to the massivemode in the Goldstone model (3).In this work, we measure the amplitude modes in a spin-1

Bose–Einstein condensate (BEC) through measurements ofvery low amplitude excitations from the GS. The results show aquantitative agreement with gapped excitation theory (10–12)and provide a platform to probe the amplitude excitation,which plays a crucial role in the KZM in spinor condensates(11, 13–15). Although in the thermodynamic limit the ampli-tude mode energy gap goes to zero at the quantum critical point(QCP), a small size-dependent gap persists for finite-size sys-tems (12). Measurements of the energy gap near the QCP arechallenging; however, our results are consistent with a smallnonzero gap. Furthermore, by using a very slow, optimizedmagnetic field ramp, we demonstrate an adiabatic quench acrossthe QCP. Such adiabatic quenches in finite-sized systems un-derlie proposals for generating massively entangled spin statesincluding Dicke states (12) and are fundamental to the ideas ofadiabatic quantum computation (16).The experiments use a tightly confined 87Rb BEC with

N = 40,000 atoms in optical traps such that spin domain for-mation is energetically suppressed. The Hamiltonian describing

Significance

Symmetry-breaking phase transitions play important roles inmany areas of physics, including cosmology, particle physics,and condensed matter. The freezing of water provides a fa-miliar example: The translational and rotational symmetries ofwater are reduced upon crystallization. In this work, we in-vestigate symmetry-breaking phase transitions of the magneticproperties of an ultracold atomic gas in the quantum regime.We measure the excitations of the quantum magnets in dif-ferent phases and show that the excitation energy (gap) re-mains finite at the phase transition. We exploit the nonzerogap to demonstrate an adiabatic (reversible) quench across thephase transition. Adiabatic quantum quenches underlie pro-posals for generating massively entangled spin states and arefundamental to the ideas of adiabatic quantum computation.

Author contributions: T.M.H. and M.S.C. designed research; T.M.H. performed re-search; T.M.H. analyzed data; T.M.H., H.M.B., M.J.B., M.A., B.A.R., and M.S.C. wrotethe paper; M.S.C. conceived the study and supervised the research; T.M.H. conceivedthe study, developed essential theory, and carried out the simulations.; M.A. and B.A.R.assisted data taking; and H.M.B. and M.J.B. developed essential theory and carried outthe simulations.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Freely available online through the PNAS open access option.1To whom correspondence should be addressed. Email: mchapman@gatech.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1600267113/-/DCSupplemental.

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this spin system in a bias magnetic field B along the z axis is(17–21)

H =~cS2 − q�Qz −N

��2, [1]

where S2 is the total collective spin-1 operator and Qz is pro-portional to the spin-1 quadrupole moment, Qzz. The coefficient~c is the collisional spin interaction energy per particle inte-grated over the condensate and quadratic Zeeman energy perparticle q= qzB2 with qz = 72 Hz/G2 (hereafter, h= 1). The lon-gitudinal magnetization hSzi is a constant of the motion (hSzi= 0for these experiments); hence the first-order linear Zeemanenergy pSz with p∝B can be ignored. The spin-1 coherent statescan be represented on the surface of a unit sphere shown in Fig.1B with axes fS⊥,Q⊥,Qzg, where the expectation value of trans-verse spin is S2⊥ = S2x + S2y , Q⊥ is the transverse off-diagonalnematic moment Q2

⊥ =Q2xz +Q2

yz, and Qz = 2ρ0 − 1, where ρ0 isthe fractional population in the jF = 1,mF = 0i state. In thisrepresentation, the coherent dynamics evolve along the con-stant energy contours of H= ð1=2ÞcS2⊥ − ð1=2ÞqðQz − 1Þ, wherec= 2N~c (22–24) (red and green orbits in Fig. 1B). The phasespace for the single-mode spin-1 condensate is similar to thatfor the Lipkin–Meshkov–Glick (LMG) model (25), which inturn describes the infinite coordination number limit of theXY model or quantum Ising model (26). The dynamics of theQPT in these zero-dimensional quantum systems have beenexplored theoretically and experimental realizations include thedouble-well Bose–Hubbard (27, 28), pseudospin-1/2 BEC (29, 30),and many-atom cavity quantum electrodynamics systems (31).

In the mean-field (large atom number) limit, quantum fluc-tuations can be ignored and the wavefunction for each spin state,mF = 0, ± 1, can be represented as a complex vector with com-ponents, ψ0,±1 =

ffiffiffiffiffiffiffiffiffiffiρ0,±1

p expðiθ0,±1Þ. Using Bogoliubov analysis(10) and mean-field theory (24, 32), the energy gap of the am-plitude mode in the P phase and the BA phase in the long-wavelength limit corresponds to the oscillation frequency ofsmall excitations in ρ0 from the GS

ΔP ≡ fP = 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqðq+ 2cÞ

p, ΔBA ≡ fBA = 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 − q2=4

q. [2]

Here the energy gap is ΔE (≡ΔP and ΔBA) and coherent oscilla-tion frequency is f (≡fP and fBA). Although these relations show avanishing gap at the QCP, quantum fluctuations due to finiteatom number result in a nonzero gap. In the quantum theory,the energy gap can be exactly calculated from the eigenenergyvalues of the Hamiltonian in Eq. 1 (Supporting Information). Fig.1C shows the energy gap between the GS and the first excitedstate with a small nonzero gap at the QCP as a result of a finiteatom number. Fig. 1D shows the relation of energy gap at theQCP to the atom number in condensates ranging from 101 to 105atoms, which scales as ΔE ∝N−1=3 (12). The energy gap curvecompares well to the oscillation frequencies of GS spinor popula-tion ρ0 obtained from quantum simulations (Supporting Informa-tion) for a broad range of atom numbers (red circles in Fig. 1D).The equivalence relation between the energy gap and the coherentoscillation (32) frequency in Eq. 2 is a general statement connect-ing the amplitude modes to the observable dynamics and is key tothis study.

Energy Gap MeasurementTo characterize the energy gap ΔE, we measure coherent dy-namics for states initialized close to the GS (Fig. 1B) for differentvalues of q=jcj ranging from 0.1 to 3 and fit the measurements tosinusoidal functions to determine the oscillation frequencies(Supporting Information). For each q=jcj value, several measure-ments of the population ρ0 are made for a series of initial statesapproaching the GS as illustrated in Fig. 2A. The GS populationρ0,GS can be obtained by minimizing the spinor energy (SupportingInformation) (24)

ρ0,GS = 1  ðPÞ, ρ0,GS = 1=2+ q=ð4jcjÞðBAÞ. [3]

The oscillation amplitude of ρ0 has a lower limit given by theHeisenberg standard quantum limit (SQL=N−1=2) projectedonto the ρ0 axis (∝Qz axis in Fig. 1B) (21); hence the best esti-mate of the energy gap is obtained from the measurement withthe lowest observable oscillation amplitude. An alternate method todetermine the energy gap for states centered on the pole is tomeasure the oscillations of the transverse spin fluctuations, ΔS⊥.Although this method requires many more data because the signalis in the fluctuations instead of the mean value, it provides highercontrast for states localized at the pole. Measurements obtainedwith this technique at the QCP are shown in Fig. 2B for a stateprepared in the polar GS (Supporting Information).The results of the energy gap measurements are shown in Fig.

2C for both methods. Overall, the measurements capture thecharacteristics of energy gap predicted by gapped excitationtheory for a spin-1 BEC (10–12). In the P phase, the energy gapdata show a good agreement with the theoretical predictionwithin the uncertainty of the measurements. In the BA phase,the measured gap data are also in reasonable agreement with thetheory; however, the measured values are 20% lower than thetheory for the smallest values of q=jcj< 1. This finding is possiblya result of small violations of the single-mode approximation orthe presence of a small thermal fraction, both of which would bemore significant in this spin interaction-dominated regime. In a

A

B

C D

Fig. 1. (A) Effective spinor potential energy V in the BA phase (q=jcj< 2), atthe QCP (q=jcj= 2), and in the P phase (q=jcj> 2). In the P phase, there aretwo gapped modes (blue lines) along the radial direction about the GS. Inthe BA phase, the GS occupies a minimum energy ring (gray circle) with onegapped mode along the radial direction (blue line) and one NG mode (redline) in the azimuthal direction. (B) The GS on the fS⊥,Q⊥,Qzg unit spheres isrepresented by a red-shaded region. Coherent orbiting (phase winding)dynamics are represented by red (green) curves and the blue curve is theseparatrix. The magenta (black) arrow represents the radio-frequency (mi-crowave) pulse used for the initial state preparation (Supporting In-formation). (C) The energy gap for 40,000 atoms (cyan curve) is calculatedfrom the eigenvalues of the quantum Hamiltonian (Supporting In-formation). (D) The energy gap at the QCP (blue solid line) calculated fromthe eigenvalues of the quantum Hamiltonian matches the GS oscillationfrequencies from simulations (red circles).

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study of an antiferromagnetic condensate, using an initial state(ρ0 = 0.5) prepared far away from the antiferromagnetic GS(ρ0,AGS = 1), slightly lower oscillation frequencies than those inthe theory were also observed (33); it was suggested that theseresulted from excess magnetization noise from the radio frequency(RF) pulse of the initial state preparation; however, this noise isnot large enough to explain the difference in our measurements.In the neighborhood of the QCP, the energy gap decreases

dramatically. As shown in Fig. 2C, Inset, the measurements are ingood agreement with the theoretical prediction in this region.For measurements at q= 2jcj, the minimum measured gap isΔE = 0.15ð1Þjcj, which is consistent with the nonzero gap pre-dicted by the quantum theory, ΔE,th = 0.165jcj; here c≈−7.5ð1ÞHz (Supporting Information). We point out, however, that thereare experimental challenges to these measurements. The initialstate is prepared in the high magnetic field GS (q=jcj= 38). Thisstate has symmetric

ffiffiffiffiN

pfluctuations in the S⊥,Q⊥ plane. When

the condensate is rapidly quenched to a lower q=jcj for the en-ergy gap measurement, this projects the condensate to slightlyexcited states of the final q=jcj Hamiltonian. The subsequentevolution of this state will have an oscillation frequency higherthan the calculated gap frequency, particularly in the region1.95≤ q=jcj≤ 2.05. We can accurately calculate this effect, andthe results are indicated by the orange-shaded region in Fig. 2C.A further complication in the measurement at the QCP is that

the value q=jcj is not truly constant during the measurement ofthe gap, but drifts to slightly higher values because of a reductionof density due to the finite lifetime of the condensate. The spininteraction energy depends on the density and atom number ascðtÞ∝ nðtÞ∝NðtÞ2=5. For these measurements, the condensatelifetime was 1.6ð1Þ s, which results in a drift of Δq=jcj= 0.05 in100 ms in the neighborhood of the QCP. The atom loss is takeninto account in the simulations, an example of which is shown inFig. 2C, and the energy gap is determined by the frequency att= 0. Despite these challenges to the measurements near theQCP, the data indicate the presence of a nonzero gap that is ofthe same size as predicted by theory.In the BA phase of a hSzi= 0 spin-1 BEC, two of the three

excitation modes (Supporting Information) are massless NGmodes that appear due to broken global symmetries. The thirdmode is a massive amplitude mode with a dispersion relation:E2ðk0Þ=Δ2

E + ð2cZ=mÞ2k20 with k0 being the wavenumber and mbeing the atomic mass (10, 11). The energy gap ΔE is equivalent

to the rest mass energy of the quasiparticle corresponding to theexcitation mode. Our experiments are in the long-wavelengthlimit in which the wave vector approaches zero, k0 → 0. Themassive amplitude mode that appears when a global symmetry isbroken has properties analogous to those of the Higgs mode,which relates to the amplitude fluctuation of the order param-eter of the phase transition. Such a Higgs-like mode has beenobserved as a collective excitation in the superfluid/Mott in-sulator transition as an amplitude fluctuation of a complex orderparameter (34), in the XY model of antiferromagnetic materialsas an amplitude fluctuation of the spin vector (35), in super-conducting systems (36–39), and here as the amplitude mode ofthe spin-1 BEC in the BA phase.

Adiabatic QPTIn the thermodynamic (N→∞) limit, the vanishing gap at theQPT prohibits adiabatic crossing between phases and gives riseto excitations characterized by the Kibble–Zurek mechanism(KZM). However, the opening of the gap at the QCP due tofinite-size effects makes it possible, in principle, to cross the QCPadiabatically using a carefully tailored ramp from q � 2jcj toq< 2jcj, while remaining in the GS of the Hamiltonian. Recently,adiabaticity in sodium spin-1 condensates has been studied (40);however, these experiments were performed using condensateswith nonzero longitudinal magnetization (hSzi≠ 0) that do nothave a QCP. Here, we focus on the very challenging case of thesmall energy gap at the QCP.Due to the small size of the gap, the ramp in q needs to be very

slow in the region of 2jcj to maintain adiabaticity. To allowlonger ramps, we used a single-focus dipole trap in which thecondensate lifetime is 15–19 s. To determine the optimal ramp,we performed simulations using measured values of the traplifetime, the atom number, and the spin interaction energy. Theramp is determined from a piecewise optimization of the Lan-dau–Zener adiabaticity parameter ðdΔE=dtÞð1=Δ2

EÞ (41–44) andincludes the effects of atom loss on c (Supporting Information).The simulations (45, 46) show that it is possible to adiabaticallycross the phase transition in ∼35 s, starting with a condensateinitially containing 40,000 atoms; here we use an adiabatic in-variant to determine the condition for adiabaticity (44, 47).The experiment starts with atoms at the GS in the polar phase at

a high magnetic field, q=jcj= 140. Then, the magnetic field isramped through the QCP to q=jcj= 1.33 in 35 s along the trajectory

A B C

Fig. 2. Energy gap measurements. (A) Coherent oscillation data (red circles) obtained at q=jcj= 2.03. In clockwise order, the oscillation amplitude decreases asthe initial state is prepared closer to the GS. Each data point is an average of 10 measurements and the data are fitted to a sinusoidal function with a varyingfrequency (solid line). (B) The time evolution of ΔS⊥ data (blue squares) at the QCP (q=jcj= 2) is fitted to a sinusoidal function (solid line). Each data point is thenoise of 45 measurements. The corresponding simulation is represented by an orange curve with the shaded region being q=jcj= 2± 0.005. (C) The energy gapΔE for different q=jcj values is obtained from the frequency fits of coherent oscillation data. Circles (triangles) are obtained from an average of 10 (or 3)measurements of ρ0 coherent oscillation, and the blue square is the frequency fit of ΔS⊥ dynamics. The theoretical energy gap is represented by the purplecurve. Inset shows the region around the QCP with the shaded orange region being the energy gap for an imperfect initialization of the GS (main text). Thedashed line shows the theoretical energy gap when the initial population ρ0 is about 0.05 away from the actual GS, reflecting an oscillation amplitude of 0.05.

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represented by the green line in Fig. 3A. The measured evolution ofthe population ρ0 is shown in the same graph and compared withthat predicted by the simulation. The data show excellent agree-ment with the theoretical values for the evolving GS populationρ0,GS (Eq. 3), which provides a strong indication of adiabaticity.There are about 9,000 atoms remaining after the adiabatic ramp.

The theoretical value of the GS population and uncertainty isρ0,GS = 0.833± 0.0047, where the uncertainty is the SQL for 9,000atoms, projected onto the ρ0 axis (∝Qz axis in Fig. 1B) (SupportingInformation). Immediately after the adiabatic ramp (t≈ 36 s), themeasured mean population and fluctuations are ρ0 = 0.830± 0.007,which are very close to the theoretical values and further indicateadiabiticity. Following the adiabatic ramp, the ratio q=jcj= 1.33 isheld constant for 2 s to verify that the system remains in the GS. Asshown in Fig. 3B, the mean value of ρ0 stays close to the theoreticalvalue ρ0,GS. In Fig. 3C, the variance Δρ20 is plotted. Although themeasurements of Δρ20 (red circles) tend above the theoretical SQLsquared (dashed line) after holding, atom loss increases fluctuationsin the spin populations (assuming uncorrelated losses) to the levelshown in the green-shaded region (Supporting Information).For comparison, in Fig. 3 D and E we show data from non-

adiabatic ramps from q=jcj= 140→ 1.33. In Fig. 3D, a 1-s linearramp is used, whereas in Fig. 3E, a 28-s ramp is used. In both cases,the spin population ρ0 does not follow the theoretical GS pop-ulation during the ramp and the variance Δρ20 grows dramatically.The fluctuations at the end of the 28-s ramp are compared withthose from the adiabatic ramp in Fig. 3C (blue squares), and it isclear that the nonadiabaticity gives rise to increased fluctuations.Adiabatically crossing the QPT in a spin-1 zero magnetiza-

tion condensate is predicted to generate massively entangledspin states (12). Broadly speaking, this is an example of thefundamental principle underlying adiabatic quantum comput-ing, in which the initial, simple GS is transformed into a highlyentangled final GS by tuning the Hamiltonian adiabaticallythrough a QCP. This final GS of the Hamiltonian is a solutionto a computation problem (16). In our case, the final state for aramp to q= 0 is predicted to be the Dicke state jS=N, Sz = 0i.In this study, we stop the adiabatic ramp at q=jcj= 1.33. The

entanglement of the GS at this q=jcj can be calculated as in ref. 12,ξ= ðhS2x i+ hS2y iÞN=ð1+ 4hðΔSzÞ2iN2Þ. The uncertainty in trans-verse magnetization is hS2x i+ hS2y i≈ 1− ð2ρ0 − 1Þ2. In the idealcase, the longitudinal magnetization is zero and conservedhðΔSzÞ2i→ 0, and the expected entanglement is ξ= 0.56N or roughly5,000 atoms are entangled out of 9,000 atoms at the end of the adi-abatic ramp. However, atom loss induces noise in the magnetization,ΔSz ≈ 0.5%, in our experiment. This small magnetization noise re-duces the entanglement to ξ< 1 atom.In summary, we have explored the amplitude mode in small spin-1

condensates. The energy gap measurements show evidence of anonzero gap at the QCP arising from finite-size effects, and usinga carefully tailored slow ramp of the Hamiltonian parameters, wehave adiabatically crossed the QCP with no apparent excitation ofthe system. We hope that this work stimulates similar investigationsin related many-body systems, and in particular, we anticipate thatthe results of this study could directly inform investigations indouble-well Bose–Josephson junction systems, (pseudo)spin-1/2interacting systems (48, 49), and the Lipkin–Meshkov–Glick (LMG)model (43, 50), which share similar Hamiltonians.

Materials and MethodsThe experiment is carried out using small condensates of 40,000 atoms in theF = 1 hyperfine GS of 87Rb. In the energy gap experiment, atoms are confined ina spherical optical dipole force trap with trap frequencies ∼ 2π ×160  Hz, formedby crossing the focus of a 10.6-μmwavelength laser with an 850-nm wavelengthlaser. This tight confinement ensures that the condensate is well described by thesingle-mode approximation (SMA), such that the spin dynamics can be consid-ered separately from the spatial dynamics (17–19). The spin interaction energyc≈−7.5ð1Þ Hz and trap lifetime is ≈ 1.6ð1Þ s. The spin populations of the con-densate are measured by releasing the trap and allowing the atoms to expand ina Stern–Gerlach magnetic field gradient to separate the mF spin components.The atoms are probed for 200 μs with three pairs of counter-propagating or-thogonal laser beams, and the fluorescence signal collected by a CCD camera isused to determine the number of atoms in each spin component.

ACKNOWLEDGMENTS. We thank K. Wiesenfeld, F. Robicheaux, T. Li, andT.A.B. Kennedy for useful discussions. The authors acknowledge supportfrom National Science Foundation Grant PHY-1506294.

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Fig. 3. Adiabatic and nonadiabatic dynamics. (A–C) Adiabatic dynamics of population ρ0 and the uncertainty Δρ0 (main text). (D and E) Nonadiabatic dy-namics of population ρ0 of 1-s and 28-s linear ramp from q=jcj= 140→ 1.33. Red circles (blue squares) are the adiabatic (nonadiabatic) measurements, gray-shaded regions are the theoretical ρ0,GS values, green arrow lines represent the ramp of q=jcj (vertical axis on the right), and cyan curves represent thedynamics simulation of ρ0 with the corresponding q=jcj ramp. Each adiabatic data point is an average of 3 measurements and each nonadiabatic data point isan average of 15 measurements.

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